In addition, if \(w_0 = 0\) , the system is in the chiral limit and \(\langle x_{\pm } \rangle \)  is almost a linear function of \(k\) , but becomes quite flat when \(k \sim 0.5\)  if the system is far away from the chiral limit (\(w_0 \lesssim w_1\) ) [1]} (also see Fig. 1 in Ref. [2]}). This behavior is also qualitatively reproduced in our constructed \(H_{eff}^{\mathbf {K}}\) , as shown in Fig. REF . For comparison, Fig. REF  also shows the winding of the phase \(\langle x_{\pm } \rangle \)  obtained from the BM model, with the values of the parameters \(v_F\) , \(w_0\) , and \(w_1\)  taken from Tab. REF . It is found that the curve for the BM model is straighter, suggesting that the terms neglected in the BM model but present in \(H_{eff}^{\mathbf {K}}\) , drive the system further away from the chiral limit. Moreover, the curve for the model in Ref. [3]} is straighter than the one for the model in Ref. [4]}, and thus consistent with the former model being closer to the chiral limit than the latter.
